Applying the above dimension reduction, the transformed device state matrix \(\mathbf {S}\)  in the low-dimensional space can be adopted for active device detection. Since the number of BS antennas in 6G wireless networks is huge, such a dimension reduction can decrease the computational complexity substantially during the recovery of the device state matrix. Importantly, \(\mathbf {S}\)  has the same distribution parameters with \(\mathbf {X}\) , a feature that will be exploited for improving detection performance in this paper. Different from this paper, the work in [1]} mainly applied the full column rank property of transformed device state matrix \(\mathbf {S}\)  after the dimension reduction.
Note that the result in (REF ) is a high signal-to-noise ratio (SNR) characterization of the received signal. Such a high-SNR approximation is a reasonable assumption in 6G wireless networks with massive connectivity due to the limited interference.
